96 The Basics of financial economeTrics
(dropping the subscript ε for the error term), we can then deflate the terms
in the bivariate linear regression given by equation (4.22) by the assumed
known standard deviation of the error term as follows:
yxt
tt
t
tt
σ tβ
σβ
σε
σ
=
+
+
(^01)
1
(4.23)
We have transformed all the variables in the bivariate regression, includ-
ing the original error term. It can be demonstrated that the regression with
the transformed variables as shown in equation (4.23) no longer has hetero-
scedasticity. That is, the variance of the error term in equation (4.23), εt/σt,
is homoscedastic.
Equation (4.23) can be estimated using ordinary least squares by simply
adjusting the table of observations so that the variables are deflated by the
known σt. When this is done, the estimates are referred to as weighted least
squares estimators.
We simplified the illustration by assuming that the variance of the error
term is known. Obviously, this is an extremely aggressive assumption. In
practice, the true value for the variance of the error term is unknown. Other
less aggressive assumptions are made but nonetheless are still assumptions.
For example, the variance of the error term can be assumed to be propor-
tional to one of the values of the independent variables. In any case, the
WLS estimator requires that we make some assumption about the variance
of the error term and then transform the value of the variables accordingly
in order to apply the WLS technique.
Absence of Autocorrelation of the Residuals
Assumption 3 is that there is no correlation between the residual terms.
Simply put, this means that there should not be any statistically significant
correlation between adjacent residuals. In time series analysis, this means no
significant correlation between two consecutive time periods.
The correlation of the residuals is critical from the point of view of
estimation. Autocorrelation of residuals is quite common in financial data
where there are quantities that are time series. A time series is said to be
autocorrelated if each term is correlated with its predecessor so that the
variance of each term is partially explained by regressing each term on its
predecessor.
Autocorrelation, which is also referred to as serial correlation and
lagged correlation in time series analysis, like any correlation, can range
from −1 to +1. Its computation is straightforward since it is simply a